Complementarity, polarity and triality in non-smooth, non-convex and non-conservative Hamilton systems

This paper presents a uni­ ed critical-point theory in non-smooth, non-convex and dissipative Hamilton systems. The canonical dual/polar transformation methods and the associated bi-duality and triality theories proposed recently in non-convex variational problems are generalized into fully nonlinear dissipative dynamical systems governed by non-smooth constitutive laws and boundary conditions. It is shown that, by this method, non-smooth and non-convex Hamilton systems can be reformulated into certain smooth dual, complementary and polar variational problems. Based on a newly proposed polar Hamiltonian, a nice bipolarity variational principle is established for three-dimensional non-smooth elastodynamical systems, and a potentially powerful complementary variational principle can be used for solving unilateral variational inequality problems governed by non-smooth boundary conditions.